An indefinite integral is the function whose derivative you were given, plus a constant. The power rule says to add one to the power and divide by the new power.
This lesson is part of SPM Additional Mathematics integration. It needs differentiation of powers, so review that first if your derivative rules are shaky.
How does the power rule work?
For any n except −1, ∫xⁿ dx = xⁿ⁺¹ ÷ (n + 1) + c. A constant multiplier stays in front, and terms are integrated one by one.
Integrate 6x² − 4x + 5:
∫(6x² − 4x + 5) dx = 6x³ ÷ 3 − 4x² ÷ 2 + 5x + c = 2x³ − 2x² + 5x + c.
The last term, 5, is 5x⁰, so it becomes 5x¹ ÷ 1 = 5x.
Worked example: finding the constant
A curve has gradient function dy/dx = 3x² − 4x and passes through (2, 5). Find its equation.
Integrate: y = x³ − 2x² + c.
Substitute the point x = 2, y = 5: 8 − 8 + c = 5, so c = 5.
The curve is y = x³ − 2x² + 5. Check: differentiating gives 3x² − 4x, which is the gradient given.
Negative powers and brackets
Rewrite before integrating. For 1 ÷ x², write x⁻²:
∫x⁻² dx = x⁻¹ ÷ (−1) + c = −1/x + c.
For a product, expand first. ∫(x + 3)(x − 1) dx = ∫(x² + 2x − 3) dx = x³/3 + x² − 3x + c.
The mistake that costs marks
The common slip is to differentiate by habit, turning x³ into 3x² instead of x⁴ ÷ 4. The two rules look like mirror images, so the direction is easy to swap under pressure.
| Step | Wrong | Right |
|---|---|---|
| ∫x³ dx | 3x² | x⁴ ÷ 4 + c |
| ∫4x dx | 4 | 2x² + c |
| ∫x⁻² dx | −2x⁻³ | −x⁻¹ + c |
| Constant | (omitted) | + c written |
A quick test for the direction: integrating raises the power, differentiating lowers it. If your power went down, you differentiated.
Check yourself
Given dy/dx = 4x³ − 6x + 1 and y = 7 when x = 1, find y in terms of x.
Answer
Integrate: y = x⁴ − 3x² + x + c.
Substitute x = 1, y = 7: 1 − 3 + 1 + c = 7, so −1 + c = 7 and c = 8.
So y = x⁴ − 3x² + x + 8.
Check: dy/dx = 4x³ − 6x + 1, and at x = 1, y = 1 − 3 + 1 + 8 = 7.
What to study next
Definite integrals put limits where the constant was. Continue with evaluating definite integrals, then test the whole chapter with the integration practice set.
You can try your own polynomials in the polynomial integration and area explorer. For a teacher to go through your working, see online one-to-one Additional Mathematics tuition.